Answer to Question #134788 in Abstract Algebra for Swathy

Question

Check whether x^5+9x^4+12x^2+6 is reducible over Q.

Answer 1

Solution: We can check whether $x^5+9x^4+12x^2+6$ is reducible over Q or not by Eisenstein's criterion.

Let $p=3$ (a prime number). We know that $a_n$ is the coefficient of highest power. Now, we notice that $3 \nmid a_n=1$, but since $3|9, 3|12, 3|6$ and $3|0$, i.e., $3$ divides all the other coefficients of given polynomial.

We also notice that $3^2=9 \nmid 6=a_0$

Thus, Eisenstein's criterion is satisfied and so, $x^5+9x^4+12x^2+6$ is irreducible over Q[x].

Answer: $x^5+9x^4+12x^2+6$ is irreducible over Q[x].